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C++ API

: d = eigs (A)

: d = eigs (A, k)

: d = eigs (A, k, sigma)

: d = eigs (A, k, sigma, opts)

: d = eigs (A, B)

: d = eigs (A, B, k)

: d = eigs (A, B, k, sigma)

: d = eigs (A, B, k, sigma, opts)

: d = eigs (af, n)

: d = eigs (af, n, B)

: d = eigs (af, n, k)

: d = eigs (af, n, B, k)

: d = eigs (af, n, k, sigma)

: d = eigs (af, n, B, k, sigma)

: d = eigs (af, n, k, sigma, opts)

: d = eigs (af, n, B, k, sigma, opts)

: [V, d] = eigs (A, …)

: [V, d] = eigs (af, n, …)

: [V, d, flag] = eigs (A, …)

: [V, d, flag] = eigs (af, n, …)

Calculate a limited number of eigenvalues and eigenvectors of A, based on a selection criteria.

The number of eigenvalues and eigenvectors to calculate is given by k and defaults to 6.

By default, eigs solve the equation where is the corresponding eigenvector. If given the positive definite matrix B then eigs solves the general eigenvalue equation

The argument sigma determines which eigenvalues are returned. sigma can be either a scalar or a string. When sigma is a scalar, the k eigenvalues closest to sigma are returned. If sigma is a string, it must have one of the following values.

"lm"

Largest Magnitude (default).

"sm"

Smallest Magnitude.

"la"

Largest Algebraic (valid only for real symmetric problems).

"sa"

Smallest Algebraic (valid only for real symmetric problems).

"be"

Both Ends, with one more from the high-end if k is odd (valid only for real symmetric problems).

"lr"

Largest Real part (valid only for complex or unsymmetric problems).

"sr"

Smallest Real part (valid only for complex or unsymmetric problems).

"li"

Largest Imaginary part (valid only for complex or unsymmetric problems).

"si"

Smallest Imaginary part (valid only for complex or unsymmetric problems).

If opts is given, it is a structure defining possible options that eigs should use. The fields of the opts structure are:

issym

If af is given, then flags whether the function af defines a symmetric problem. It is ignored if A is given. The default is false.

isreal

If af is given, then flags whether the function af defines a real problem. It is ignored if A is given. The default is true.

tol

Defines the required convergence tolerance, calculated as tol * norm (A). The default is eps.

maxit

The maximum number of iterations. The default is 300.

p

The number of Lanzcos basis vectors to use. More vectors will result in faster convergence, but a greater use of memory. The optimal value of p is problem dependent and should be in the range k to n. The default value is 2 * k.

v0

The starting vector for the algorithm. An initial vector close to the final vector will speed up convergence. The default is for ARPACK to randomly generate a starting vector. If specified, v0 must be an n-by-1 vector where n = rows (A)

disp

The level of diagnostic printout (0|1|2). If disp is 0 then diagnostics are disabled. The default value is 0.

cholB

Flag if chol (B) is passed rather than B. The default is false.

permB

The permutation vector of the Cholesky factorization of B if cholB is true. That is chol (B(permB, permB)). The default is 1:n.

It is also possible to represent A by a function denoted af. af must be followed by a scalar argument n defining the length of the vector argument accepted by af. af can be a function handle, an inline function, or a string. When af is a string it holds the name of the function to use.

af is a function of the form y = af (x) where the required return value of af is determined by the value of sigma. The four possible forms are

A * x

if sigma is not given or is a string other than "sm".

A \ x

if sigma is 0 or "sm".

(A - sigma * I) \ x

for the standard eigenvalue problem, where I is the identity matrix of the same size as A.

(A - sigma * B) \ x

for the general eigenvalue problem.

The return arguments of eigs depend on the number of return arguments requested. With a single return argument, a vector d of length k is returned containing the k eigenvalues that have been found. With two return arguments, V is a n-by-k matrix whose columns are the k eigenvectors corresponding to the returned eigenvalues. The eigenvalues themselves are returned in d in the form of a n-by-k matrix, where the elements on the diagonal are the eigenvalues.

Given a third return argument flag, eigs returns the status of the convergence. If flag is 0 then all eigenvalues have converged. Any other value indicates a failure to converge.

This function is based on the ARPACK package, written by R. Lehoucq, K. Maschhoff, D. Sorensen, and C. Yang. For more information see http://www.caam.rice.edu/software/ARPACK/.

See also: eig, svds.

Package: octave